Find dy/dx x^2y^2=36

Problem

x2*y2=36

Solution

1. Apply implicit differentiation by taking the derivative of both sides of the equation with respect to x

(d(x2)*y2)/d(x)=d(36)/d(x)

2. Use the product rule on the left side, where (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x) and remember to use the chain rule for terms involving y

x2d(y2)/d(x)+y2d(x2)/d(x)=0

3. Differentiate the terms, treating y as a function of x

x2*(2*yd(y)/d(x))+y2*(2*x)=0

4. Simplify the expression to prepare for solving for d(y)/d(x)

2*x2*yd(y)/d(x)+2*x*y2=0

5. Isolate the term containing d(y)/d(x) by subtracting 2*x*y2 from both sides.

2*x2*yd(y)/d(x)=−2*x*y2

6. Solve for dy/dx by dividing both sides by 2*x2*y

d(y)/d(x)=(−2*x*y2)/(2*x2*y)

7. Simplify the fraction by canceling common factors of 2 x and y

d(y)/d(x)=−y/x

Final Answer

d(y)/d(x)=−y/x

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